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Quantile Regression
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The intuition Hypothetical Distributions
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The intuition OLS Regression Results
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The intuition Quantile Regression Results
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An alternative approach Logistic Regression Models
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Examples Five ideas from your (or your friends’) research where this approach might be useful.
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Some examples Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf)
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Some examples Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf)
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Some examples
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Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf) Some examples
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Image from Bitler et al. 2006 AER paper.
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Some examples
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Some more examples
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Some examples Pronghorn densities (y) by shrub canopy cover (X) on n = 28 winter ranges (data from Cook and Irwin 1985) and 0.90, 0.75, 0.50, 0.25, and 0.10 regression quantile estimates (solid lines) and least squares regression estimate (dashed line) for the model y = b0 + b1X +e. (From Cade and Noon, 2003).
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Some examples Quantile regression was used to estimate changes in Lahontan cutthroat trout density (y) as a function of the ratio of stream width to depth (X) for 7 years and 13 streams in the eastern Lahontan basin of the western US. A scatterplot of n = 71 observations of stream width:depth and trout densities with 0.95, 0.75, 0.50, 0.25, and 0.05 quantile (solid lines) and least squares regression (dashed line) estimates for the model ln y = b0 + b1X +e. From Cade and Noon, 2003.
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Technical intuitions Image from Pindyck and Rubinfield (Econometric models and economic forecasts)
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Formulae (OLS)
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Formulae (LAD)
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Formulae (LAD vs OLS)
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Formulae (LAD at ≠.5)
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Negative residualsPositive residuals
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Technical (semi) intuitions Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf)
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Why we might care
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Skewed Distributions
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Issues Small samples –Guidelines: The 30 observations rule? (Chernozhukov) Suitable dependent variables –Does your metric make sense? Accessibility –(Relatively) new outside of economics –Solution: Find a friend in economics? –More difficult with thornier data (categorical DV’s, panel data, etc)
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Issues Cluster robust standard errors –Solutions: Bootstrapping se’s Sandwich estimators (see stata code online) Thinking about effects –Effects on the distribution –Rank preservation assumptions Distribution of Y not of X
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